Simultaneous Production and Distribution of Industrial Gas Supply-Chains Pablo A. Marchetti 1 , Vijay Gupta 1 , Ignacio E. Grossmann 1† , Lauren Cook 2 , Pierre-Marie Valton 3 , Tejinder Singh 2 , Tong Li 2 , and Jean André 3 1 Dept. of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213 2 American Air Liquide Inc., Delaware Research and Technology Center, Newark, DE 19702 3 Air Liquide, Paris Saclay R&D Center, 78350 Jouy-en-Josas, France Abstract In this paper, we propose a multi-period mixed-integer linear programming model for optimal enterprise- level planning of industrial gas operations. The objective is to minimize the total cost of production and distribution of liquid products by coordinating production decisions at multiple plants and distribution decisions at multiple depots. Production decisions include production modes and rates that determine power consumption. Distribution decisions involve source, destination, quantity, route, and time of each truck delivery. The selection of routes is a critical factor of the distribution cost. The main goal of this contribution is to assess the benefits of optimal coordination of production and distribution. The proposed methodology has been tested on small, medium, and large size examples. The results show that significant benefits can be obtained with higher coordination among plants/depots in order to fulfill a common set of shared customer demands. The application to real industrial size test cases is also discussed. Keywords Supply-chain optimization, Industrial gases, Production planning, Inventory routing problem, Multi- period model, Mixed-integer linear programming † Corresponding author. E-mail address: [email protected]1
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Simultaneous Production and Distribution of Industrial Gas Supply-Chains
Pablo A. Marchetti1, Vijay Gupta1, Ignacio E. Grossmann1†, Lauren Cook2, Pierre-Marie Valton3, Tejinder Singh2, Tong Li2, and Jean André3
1Dept. of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213 2American Air Liquide Inc., Delaware Research and Technology Center, Newark, DE 19702
3Air Liquide, Paris Saclay R&D Center, 78350 Jouy-en-Josas, France
Abstract
In this paper, we propose a multi-period mixed-integer linear programming model for optimal enterprise-
level planning of industrial gas operations. The objective is to minimize the total cost of production and
distribution of liquid products by coordinating production decisions at multiple plants and distribution
decisions at multiple depots. Production decisions include production modes and rates that determine
power consumption. Distribution decisions involve source, destination, quantity, route, and time of each
truck delivery. The selection of routes is a critical factor of the distribution cost. The main goal of this
contribution is to assess the benefits of optimal coordination of production and distribution. The proposed
methodology has been tested on small, medium, and large size examples. The results show that significant
benefits can be obtained with higher coordination among plants/depots in order to fulfill a common set of
shared customer demands. The application to real industrial size test cases is also discussed.
Keywords
Supply-chain optimization, Industrial gases, Production planning, Inventory routing problem, Multi-
Moreover, additional constraints limiting the total liquid production for a given production mode m at
plant p can be specified by Eqn (5), where the parameters ,pm iλα and ,pm λπ are the coefficients and
upper bound, respectively, for a linear combination of the production rates of every product i. The set
LIMm stands for the limits of the feasible region of production mode m, where each λ is associated to a
limiting hyperplane. Notice that in each mode m we assume that the plants are flexible enough to operate
anywhere within the limits given by Eqns (4) and (5).
, , , , , ,pm
pm i pmi t pmt pm m pti I
W B LIM m M p P t Tλ λα π λ∈
≤ ∀ ∈ ∈ ∈ ∈∑ (5)
The power consumption of plant p in time period t is given by Eqn (6), where the parameter usppmi is the
energy requirement per unit of product i (unit specific power) when plant p operates in production mode
m.
( ) TtPpWuspPWpt pmMm Ii
tpmipmitp ∈∈∀⋅= ∑ ∑∈ ∈
,,, (6)
11
4.1.3 Inventory constraints at plants
Storage is assumed to be available at the plants to keep the inventory of every product i∈Ip. The
continuous variable Lpit stands for the inventory level of liquid product i at plant p at the end of time
period t. Equation (7) establishes the lower and upper bounds for the level of product i, which must lie
between the minimum level (redline) and the maximum storage capacity of the facility for that product.
The minimum inventory level ensures that excess demand of over-the-fence/on-site gaseous customers
can be met using this inventory as a back-up source. Moreover, this redline ( minpitQ ) is a given parameter
that may vary over the planning horizon (not constant) based on the gaseous customer demand profile,
while the maximum limit maxpiQ is related to the physical capacity of the storage facility (a constant value)
for product i.
TtPpIiQLQ ppipitpit ∈∈∈∀≤≤ ,,maxmin (7)
The material balance constraints (8) are required to keep track of the inventory level of product i at each
time period t. In particular, the amount of product in storage at plant p is equal to the inventory of the
product at the previous time period, plus the production over time period t, minus both the total amount of
product supplied on-site ( sitetpiD , ) and the total product distributed by trucks ( truck
tpiD , ) at time t. The variables
sitetpiD , and truck
tpiD , are introduced in the next sections. Also, for the first time period the value of Lpi,t-1 is
given by the inventory level of the plant when the time horizon begins ( inipiL ).
TtPpIiDDWLL ptruck
tpisite
tpiMm
tpmittpitpiipt
∈∈∈∀−−∆+= ∑∈
− ,,,,,1,,,
(8)
Material balance constraints (8) are the main constraints that connect the production and distribution sides
of the supply chain.
4.1.4 Gaseous customer supply
As indicated by Equation (9), the amount of product distributed on-site for each time period t is defined as
the gaseous volume supplied by vaporization through the pipeline to an over-the-fence customer sitting
near the plant. This vaporization of liquid product is needed only when the gas can not be supplied from
12
the separation column because the plant is shut down. The parameter sitetpiR , is the demand forecast of the
over-the-fence customer for product i.
TtPpIiBRD pMm
pmtsite
tpisite
tpipt
∈∈∈∀
−= ∑
∈
,,1,, (9)
4.2 Distribution Side
The main distribution decisions include the amount of product being delivered from a given source, the
truck being used, and the set of customers being visited within a given time period. We assume here that
a truck performs a round-trip on each time period t. While this assumption is valid most of the time, in the
general case a driver can eventually complete two or three trips in a single shift before finishing his
working hours. To allow several short trips in a single shift, the duration of each time period may be
reduced to obtain a more accurate discretization of the time horizon. However, this increases the model
size and the computational effort required to find solutions that are accurate enough (just dividing each
time period by two duplicates the number of binary variables and constraints of the model). Besides, it is
also assumed that a single trip starts and ends at the same depot from which the truck departs. While the
inventory capacity constraints are verified only at the time interval limits, it is assumed that truck loading
and unloading tasks may occur anytime within these limits and that there is enough capacity available to
accommodate the production-distribution schedule if needed.
If multiple combined trips are needed, the problem becomes a multiple source pick-up and delivery
problem, a level of detail that is not tackled in this contribution. Multiple trucks, multiple sources and
multiple alternative routes generate a combinatorial explosion of the number of alternatives to be explored
on the distribution side of the supply-chain.
4.2.1 Selection of routes
Each truck is assigned to a fixed depot and dedicated to transport a unique kind of product. Thus, the set
of trucks k∈Kdi is defined for every depot d and product i. As shown in Figure 3, distribution by trucks is
accomplished by the following steps: (a) a truck k travels from its depot d to a valid source location (i.e., a
related plant p), (b) product i is loaded at plant p such that the truck capacity truckkU is not exceeded, (c)
truck k visits a set of customers s (one or more) and delivers the product, which is distributed in any
required proportion among them, and (d) truck k travels back to its depot. Therefore, given the depot d,
13
the plant p, and a set of customers s, the route with the shortest distance (disdps) to complete the delivery
can be calculated a-priori (pre-processed).
Figure 3. Routes are determined by combining a depot, a plant, and a customer set.
The binary variables Ykpt and ykst are introduced to indicate whether or not truck k is associated to plant p
and customer set s, respectively, at time period t. Since each truck is associated to a known depot, there is
no need to decide on the depot to be used on a given trip. Constraints (10) and (11) together indicate that
each truck k can only be assigned to a single route on each time interval t. Equation (10) represents the
fact that a truck k can be assigned to a single set of customers per time period. If the LHS is one, then the
truck k is delivering product at time t.
TtDdIiKky diSs
kstdi
∈∈∈∈∀≤∑∈
,,,1 (10)
The set Sdi stands for all the customer sets associated with routes for product i that start at depot d.
Because of Equation (10), at most one customer set s is selected for truck k to visit at each time period t.
In turn, each customer set s can include one or more customers. Since the number of possible sets s grows
very fast with the number of customers of product i (i.e., | Ci |), an effective route selection method is
required to keep the model size reasonable. Appendix A describes the route selection method used herein,
which is based on the idea of enumerating all feasible routes, sorting them using an economic criterion,
and selecting the most appropriate ones while guaranteeing a minimum number of routes for each
customer. Practical sorting criteria are either the route distance or an estimation of the cost per volume
sourced for the route.
Given Equation (10), Equation (11) establishes that a sourcing plant is required if and only if truck k is
delivering product at time t.
TtDdIiKkyY diSs
kstPp
kptdidi
∈∈∈∈∀= ∑∑∈∈
,,, (11)
14
The set Pdi includes all the plants that are authorized to source product i by loading a truck from depot d.
Section 4.2.7 will further discuss possible delivery restrictions that apply when taking into account
different product grades.
4.2.2 Truck load constraints
Continuous non-negative variables Ekpt and ekst are introduced to handle the quantity of product delivered
by truck k. The variable Ekpt represents the amount of product loaded by truck k at plant p in time period t,
while the variable ekst is the amount delivered by truck k to customer set s in the same time period. Since
only one source is allowed for a given truck, constraint (12) guarantees that only the appropriate variable
Ekpt is nonzero for some p∈Pdi.
TtDdIiPpKkUYE diditruckkkptkpt ∈∈∈∈∈∀≤ ,,,, (12)
Also, constraint (13) states that the variable ekst can be nonzero only if truck k delivers to the customer set
s (ykst = 1).
TtDdIiSsKkUye diditruckkkstkst ∈∈∈∈∈∀≤ ,,,, (13)
Finally, given the aforementioned bounds for variables Ekpt and ekst, Equation (14) is needed to ensure that
the amount of product picked up at a given plant is the same one being delivered to the selected
customers, for each truck k and time period t.
TtDdIiKkEe diPp
kptSs
kstdidi
∈∈∈∈∀= ∑∑∈∈
,,, (14)
4.2.3 Plant pick-up and customer delivery amounts
Given equations (12)-(14), three additional constraints are needed to connect both sides of the supply-
chain.
On one hand, Equation (15) defines the amount of product i delivered by truck from plant p at each time t
(i.e., trucktpiD , ) as the summation of the product loaded by every truck that stops at p at that time period.
Delivery limitations established for the depots are taken into account by including the condition p∈Pdi.
15
∑ ∑∈∈ ∈
∈∈∈∀=)(:
, ,,di diPpDd
pKk
kpttruck
tpi TtPpIiED (15)
On the other hand, Eqns (16) and (17) are used to determine the total amount of product delivered to a
given customer c at time t (Dc,t). Constraint (16) ensures that the product being delivered to each customer
set s is split among the customers c∈s. To this end, the continuous variable dsct is introduced to indicate
the amount of product that customer c receives at time period t from all trucks that deliver to customer set
s at that time. Notice that the LHS of Eqn (16) is the amount of product carried by all trucks that visit
customer set s, and the RHS is the amount delivered to the customers in s. Moreover, the set Si includes
all customer sets for a given product i.
TtIiSsde i
scsct
SsDd Kkkst
di di
∈∈∈∀=∑∑ ∑∈∈∈ ∈
,,)(:
(16)
Finally, constraint (17) calculates Dc,t as the summation of all the deliveries being made to c through all
relevant sets s.
∑
∈∈
∈∈∈∀=)(:
, ,,scSs
iscttci
TtIiCcdD (17)
Figure 4 depicts the material flow represented by the material balance constraints (15), (14), (16) and (17)
and defined with the continuous variables trucktpiD , , Ekpt, ekst, dsct, and Dc,t. Also, Figure 5 shows in more
detail the interpretation of the material balance constraint (15) when multiple trucks from different depots
load product at plant P1.
16
Figure 4. Distribution side continuous variables used to represent the delivery of liquid products from
plants to customers.
Figure 5. Loading of multiple trucks at a given plant.
17
4.2.4 Route distances
The next set of constraints is needed to determine the distance traveled by a truck k when a delivery is
made at time period t. As mentioned before, given the depot d, source p, and destination s (i.e. a set of
customers) associated with each possible trip, the shortest traveling distance (disdps) can be calculated a
priori. This can be done either by enumerating all possible alternatives or using a specific TSP algorithm,
mainly because the number of customers in every customer set s is relatively small.
For each truck k∈Kdi departing from depot d at time period t, its selected route will be determined by the
specific binary variables Ykpt (p∈Pdi) and ykst (s∈Sdi) that are equal to one. However, since the information
on the route is disaggregated on these binary variables, it is not straightforward to calculate the distance
traveled by truck k.
Using the parameter disdps, Eqn (18) defines the minimum distance ( mindsdis ) required to deliver product i
to the set of customers s⊂Ci using any truck from depot d. In other words, the parameter mindsdis is the
traveling distance for the closest plant, taking into account a route with a fixed depot d and customer set s.
DdIiSsdisdis didpsPpdssdi
∈∈∈∀=∈
,,][min,
min (18)
Given the parameter mindsdis , if a source different than the closest one is used, then an additional distance
must be added in order to account for the correct delivery cost.
To this end, a non-negative continuous variable βkt is introduced, representing the distance added to mindsdis to account for a source different than the closest one (usually the default source). Constraint (19)
sets the lower bound for variable βkt based on the source and customer-set decision variables, where the
parameters δdps and max,idpδ are defined in equations (20) and (21). The parameter δdps represents the
additional distance needed between the minimum ( mindsdis ) and the complete distance (disdps), when plant
p is selected.
Furthermore, max,idpδ is the maximum distance δdps taking into account all routes associated to depot d and
product i. When Ykpt = 1 (i.e. the plant p has been selected for the truck k), the RHS of constraint (19)
becomes equivalent to βkt, and the summation on the LHS provides the adequate lower bound for the
additional distance to be considered. Otherwise, Ykpt = 0 and variable βkt can always be driven to zero
while Eqn (19) is still satisfied.
18
( ) TtDdIiKkPpYy didiktkptidpkstSs
dpsdi
∈∈∈∈∈∀+−≤∑∈
,,,,1max, βδδ (19)
with the definition of the following parameters:
DdIiPpSsdisdis dididsdpsdps ∈∈∈∈∀−= ,,,minδ (20)
DdIiPp didpsSsidpdi
∈∈∈∀=∈
,,][maxmax, δδ (21)
Finally, the distance traveled by truck k in time period t is given by the continuous variable DISkt, which is
defined in Eqn (22). The RHS includes: (a) the minimum distance required to deliver product i to
customer set s from the plant that is more conveniently located and (b) the additional distance βkt that is
needed if a different plant is selected.
TtDdIiKkydisDIS diktkstSs
dsktdi
∈∈∈∈∀+= ∑∈
,,,min β (22)
Figure 6 shows an example where a given truck k1 from depot D1 is delivering product to customers c1, c2,
and c3 (i.e., customer set s1). Two alternative routes are shown: r1 = (D1,P1,s1) and r2 = (D1,P2,s1). Thus,
the minimum distance needed to make the delivery is 11111 ,,
min, sPDsD disdis = , where P1 is the closest plant.
Moreover, the additional distance if plant P2 is selected is given by min,,,,, 11121121 sDsPDsPD disdis −=δ . If
1,, 11=tsky and 1,, 11
=tPkY then 111111 ,,
min,, sPDsDtk disdisDIS == . Otherwise, if 1,, 11
=tsky and 1,, 21=tPkY then
tksDtk disDIS ,min
,, 1111β+= and because of Equations (18)-(20) we have
1211 ,,, sPDtk δβ ≥ from which
1211 ,,, sPDtk disDIS ≥ can be derived.
19
Figure 6. Routes with alternative sources for the same depot and customer set.
4.2.5 Inventory constraints at customer sites
The inventory level at customer locations must also be tracked by the model. For each customer c∈Ci, the
level of product i inventory at the end of time t (Lct) must lie between the minimum desired level (safety
stock) and the maximum storage capacity of the tank as established by Eqn (23). Notice that the safety
stock can be given as a parameter with variations over the planning horizon based on the consumption
profile of that particular customer.
TtIiCcQLQ icctct ∈∈∈∀≤≤ ,,maxmin (23)
Constraint (24) represents the material balance constraint for the inventory of product i at each customer
location. In particular, the amount of product i in the customer storage tank in time period t is equal to the
inventory of that product at the previous time period, plus the product delivered to the customer in time
period t, Dct, less the amount of product consumed by the customer, Rct, in the same time t. For the first
time period t0, the inventory at the previous time period t – 1 will be given by the initial inventory of
product i at customer c ( inicL ).
TtIiCcRDLL ictcttcct ∈∈∈∀−+= − ,,)1( (24)
20
As an alternative to Equations (23) and (24), Appendix B presents the constraints required when the
volumes and time windows for each delivery are specified beforehand (planned deliveries).
4.2.6 Deliveries from alternative sources
Industrial gas customers usually have strict requirements on product availability bound by specific
contractual obligations. In general, the golden rule for any industrial gases provider is that a customer
must never run out of product. Thus, if the available inventory at the owned plants is not enough to fulfill
some required obligations, then the product must be provided by purchasing it from an alternative source
in order to replenish any customer inventory levels that are subject to redline conditions in a timely
manner.
In this section we indicate the changes in the mathematical model required to handle the possibility of
purchasing product from an alternative source. To this end, the set of plants P is split into two disjoint
subsets Pown and Palt, standing for the owned plants and the alternative sources (i.e. typically plants owned
by other companies). By doing this, the set P must be replaced by Pown in the Eqns (1)-(9) that model the
production side of the supply chain (see Section §4.1). However, constraints (11), (12), (14), (15), and
(18)-(21) defined in Sections §4.2.1 to §4.2.4 remain unchanged, since now the set altown PPP ∪= also
includes the alternative sources. For each additional source p∈Palt, variables Ykpt and Ekpt are also
included. Given these modifications, the total amount of product i purchased at an alternative source
p∈Palt at time period t ( trucktpiD , ) is still defined by Eqn (14). Besides, the maximum amount of product i
that can be purchased at time t is now given by the parameter purchasetpiQ , , as indicated by Eqn (25).
TtPpIiQD altp
purchasetpi
trucktpi ∈∈∈∀≤ ,,,, (25)
4.2.7 Sourcing and product grade constraints
Different grades of industrial gas products can be easily handled by the proposed method. For example,
when liquid oxygen (LOX) is considered, a distinction may be made between industrial LOX and medical
LOX, since they have different product purities. While it is possible to handle the different product grades
as different products, this approach may turn out to be over-restrictive. For example, a customer
requesting a lower grade product could also receive a higher grade, as long as the required purity
specifications are met. Given that each plant p produces a grade j = grade(p, i) for product i, let us
consider the binary relation of product grades R(j, j’) such that the demand of a customer requiring j can
be fulfilled by delivering j’. Thus, R should be a reflexive and transitive relation. Based on this relation a
21
set Jc including all product grades that can be delivered to customer c can be obtained. Consequently, the
set of plants from which product i can be sourced to a given customer set s is:
{ } IiSsJipgradePpP icscs ∈∈∀∈∈= ∈ ,),(:
A customer set s should not be considered in the model if Ps = ∅. Eqn (26) defines the set Spi, which
includes all customer sets where product i of plant p can be delivered. The definition piPpdi SSdi∈=
should be employed when Eqn (26) is used.
{ } IiPpPpSsS sipi ∈∈∀∈∈= ,: (26)
With the above definitions, constraint (27) must be added to the mathematical formulation to handle
multiple product grades for a given product i, representing the possibility to deliver products of higher
purity if available to customers who require a lower grade of the same product.
TtDdIiKkPpyY didiSs
kstkptpi
∈∈∈∈∈∀≤ ∑∈
,,,, (27)
In general, constraint (27) can be applied to restrict the selection of the customer sets s that can be sourced
from plant p, when using a truck from depot d. To this end the set Spi must be replaced by a set Spi,d, which
also takes into account the depot. For example, this situation appears when a route given by d, p, and s
exceeds a given maximum distance.
4.2.8 Tightening constraints
Valid cuts that do not eliminate integer solutions from the feasible space are added to the mathematical
model in order to improve its computational performance. The proposed cuts are intended to tighten the
LP relaxation by improving the calculation of the distribution cost.
Let ),( 21 ttcµ with t1 ≤ t2 be the summation of the product consumed by customer c in the interval from
time period t1 to time period t2, as stated by Eqn (28).
212121 :},{,),(2
1
ttTttCcRttt
ttctc ≤⊂∈∀=∑
=
µ (28)
22
Tightening constraint (29) imposes that at least one delivery must be made to each customer c within a
given interval [t1, t2]. The LHS of (29) is the number of trucks visiting all customer sets s that include c
within the proposed interval.
),()1,()(:
},{,,1
21minmax
2121
21)(:
2
1
ttQQtttt
TttIiCcy
cccc
i
t
tt Dd Kk scSskst
di di
µµ <−≤−∧≤
⊂∈∈∀≥
∑ ∑ ∑ ∑= ∈ ∈ ∈∈ (29)
The selection of the intervals for which Eqn (29) is defined is explained next. The maximum inventory
available at customer c between two consecutive replenishments is given by the expression minmaxcc QQ − .
If the product consumed between t1 and t2, i.e. ),( 21 ttcµ , is higher than this difference a delivery must be
made to customer c within [t1, t2]. This condition is therefore necessary for Eqn (29). Besides, to avoid
redundant additional constraints the condition minmax21 )1,( ccc QQtt −≤−µ is also needed. For instance,
let Eqn (29) be defined for a given interval [t1, t2]. Then, for any t3 > t2 the condition minmax
31 )1,( ccc QQtt −≤−µ does not hold because minmax2131 ),()1,( cccc QQtttt −>≥− µµ . In this way
the constraint (29) is included only for the shortest time interval starting at each time period t1. When t1 is
the first period of the time horizon, maxcQ can be replaced by the initial inventory of customer c ( ini
cL )
without loss of generality.
4.3 Objective Function
The proposed mathematical model seeks to minimize the overall cost of production and distribution for
the entire time horizon. The objective function is given by Equation (30). Equation (31) defines the
production cost for each time period t, which is given by the start-up and variable production costs of
each plant. Besides, Equation (32) sets the distribution cost at time t as the cost of all deliveries made by
every truck plus the cost of the product purchased from the alternative sources at the given time period.
We should note that we are not including inventory cost as it is normally a minor cost compared with the
production and distribution costs. However, it is clear that inventory costs can be trivially included in
(30).
23
( )∑∈
+Tt
tt DCostPCostMinimize (30)
( )∑∈
⋅∆⋅+⋅=ownPp
pttptstartpt
startptt uPWbFPCost (31)
( )∑∑∑ ∑∈∈ ∈ ∈
⋅+
⋅=
altdi Pp
trucktip
purchtip
Dd Ii Kkktkt DCDIScDCost ,,,, (32)
4.4 Modeling different levels of production-distribution coordination
The simultaneous production-distribution coordination model is given by Equations (1)-(27) and
objective function (30). This fully coordinated model is referred as model (M1). Sequential models are
derived from (M1) by decomposing the production and distribution optimization into two separate
programs that will be connected through a sequence of decisions involving both. We introduce first the
production optimization model (M2) generating the production side schedule that minimizes the total cost
of production. This model includes the constraints (1)-(9), with objective function (33).
( )∑∈Tt
tPCostMinimize (33)
Two options have been considered to set the production targets: either trucks withdrawals truck
tpiD , are
forecasted directly based on historical frequencies (M2.a) or planned deliveries are set for each customer
based on its consumption forecast, storage capacity, and historical delivery data (M2.b). Equation (34)
fixes the variable trucktpiD , for the model (M2.a). In this case, the truck withdrawal volume in each time
period t is given by the parameter withdrawtpiU , . Alternatively, when the production side model (M2.b) is
used, Equations (35) and (36) are employed to determine how much product is delivered to each customer
in order to fulfill their forecasted demands. The parameter delivttcU21,, indicates the volume of product i to be
delivered to customer c∈Ci during the time interval [t1, t2], where t1 ≤ t2. Besides, the variable pctσ is
introduced representing the product delivered from plant p to customer c at time t.
TtPpIiUD pwithdraw
tpitruck
tpi ∈∈∈∀= ,,,, (34)
24
TtPpIiD pCc
pcttruck
tpitpi
∈∈∈∀= ∑∈
,,,
, σ (35)
0:},{,,2121
2
1 ,
,,21,, >⊂∈∈∀=
∑ ∑= ∈
delivttci
delivttc
t
tt Pppct UTttIiCcU
tc
σ (36)
Finally, we introduce the distribution side optimization program (M3) generating the distribution schedule
that minimizes the total distribution cost. Constraints (7)-(9) and (11)-(27), with objective (37) are used.
We assume that the variables that handle production mode selection (Bpmt) and production rate (Wpmi,t) are
fixed taking into account a solution of a previously solved model (M2).
( )∑∈Tt
tDCostMinimize (37)
To show the potential impact of a better coordination of production and distribution decisions, we
compare the simultaneous model (M1) with the sequence (M2) → (M3), the later being to determine the
production decisions first and then observing the consequences on the availability of product before
solving the distribution model.
5. Results and Discussion
The models (M1), (M2) and (M3) were implemented in GAMS 24.1.3 and solved using the commercial
solver CPLEX 12.5.1. Computational results were obtained on an Intel Core i7-960 (3.20 GHz, 4 cores)
machine with 16 GB of RAM. All instances were solved using the parallel processing capacities of the
machine and a relative gap tolerance of 0.01, otherwise default solver setting were used. Two examples
including simultaneous production decisions at multiple plants and distribution decisions at multiple
depots are presented. Besides, the application of the proposed model to industrial size problem instances
is discussed.
5.1 Example 1
A first small test case is presented featuring two plants and two main products (LIN i.e. liquid nitrogen
and LOX i.e. liquid oxygen). A unique grade is considered for each product. The plants can be operated in
two production modes (High LIN and High LOX) with specific capacity limits. For each plant and
product, Table 2 includes the maximum rate for each production mode together with the inventory levels,
25
maximum storage capacity, and redline (minimum level). The minimum production rates are established
by a turndown ratio of 60% for plant P1 and 70% for plant P2. All product quantities are given in thousand
standard cubic feet (Mcf). Figure 7 shows the feasible production rates for each plant and production
mode. The unit specific power is 20 kWh/Mcf for every plant, product, and production mode. Besides, we
assume that both plants are initially running, and the associated start-up costs are $7,000 for plant P1 and
$4,000 for plant P2.
Table 2. Plant production and storage data for Example 1
Plant P1 P2 Unit
Product LIN LOX LIN LOX
wmax Mode Hi LIN 108 95 100 105 Mcf/h Mode Hi LOX 190 37 185 48 Inventory Initial 3,500 4,800 4,700 4,000 Mcf Maximum 9,000 6,300 8,100 7,000 Redline 3,000 2,100 2,500 1,750
Figure 7. Production rate limits for each operating mode and plant for Example 1.
There is a depot located beside each plant. Depot D1 is located at plant P1 and has 5 trucks, 3 with a trailer
for LIN and 2 with a trailer for LOX. Also, depot D2 is located at plant P2 and has 4 trucks available, 2 for
LIN and 2 for LOX. The transportation cost of trucks is 2.85 $/mile, and each trailer has a capacity of 630
Mcf. The supply-chain includes 9 customers (5 LIN customers and 4 LOX customers) to be served by
26
truck delivery. Figure 8 shows a map including all plant/depot and customer locations, which are also
indicated in Table 3. Straight line paths are used to calculate route distances. Table 4 includes the liquid
product initial inventory level, storage capacity, and redline for each customer, together with the average
consumption per day. The default source for LIN customers c1, c2, and c3 and LOX customers c6 and c7 is
plant P1. The remaining customers are associated with plant P2. Thus, as it can be observed in Figure 8,
the default source for each customer is the plant in closest proximity.
Figure 8. Map for Example 1.
Table 3. Location of plants, depots, and customers for Example 1 (miles).
A brief explanation of the results of Table 9 and Figures 9 and 10 is as follows. The sequential production
model based on truck withdrawals (M2.a) uses as its production target an estimation of the number of
full-load trucks required at each plant. Because this estimation is higher than the actual demand, some
extra production is made in addition to the amount required by the default customers. Thus, the
simultaneous distribution model (M3) is able to source some product to c2 and c7 using plant P2. However,
the volume sourced from P2 to the shared customers is still restricted by the production targets. In turn,
the best solution of the sequential model (M2.b) with production based on planned deliveries is different.
In this case most of the volume required by customers c2, c3, and c7 is sourced from plant P2. While it
30
reduces the overall production cost, it turns out that trucks from depot D1 are required to deliver the
product from plant P2, which increases the distribution cost. Finally, the fully coordinated model takes
into account both production and distribution resources to find a balanced solution that shifts most of the
demand of c2 and c7 to plant P2, without significantly penalizing the distribution cost.
The model size and computational statistics obtained by the application of the simultaneous coordination
strategy (M1) with dynamic sourcing is presented in Table 10. The remaining production-distribution
models applied to Example 1 have shown similar computational performance, with CPU times varying
between 10 and 150 s.
Figure 9. Cost comparison for the alternative levels of Production-Distribution Coordination.
Figure 10. Product sourced per plant for each multi-plant coordination strategy (Example 1).
31
Table 10. Model size and performance for the multi-plant simultaneous production-distribution model (Example 1).
Multi-plant Simultaneous Model
Binary variables 1,344 Continuous vars. 2,395 Constraints 2,916 MIP solution 63,089.45 CPU time 11.09 s Relative gap 1% Nodes 3,618
5.2 Example 2 A medium size example adapted from a real industrial size test case is presented next. Example 2 includes
three plants producing two main products (LIN and LOX). Similar to Example 1, there is a unique grade
for each product, and each plant can operate in two different production modes (High LIN and High
LOX). Production rate limits for each facility and production mode are shown in Figure 11.
Besides, the supply chain includes 3 depots and one alternative source. Depots D1 and D3 are located at
plants P1 and P3, respectively. Both have 5 trucks, 3 with a trailer for LIN and 2 with a trailer for LOX.
Depot D2 is a standalone depot located nearby plant P2. It has 4 trucks available, 2 for LIN and 2 for
LOX. The alternative source Alt1, which produces both products, is located at the north-east of depot D1
and the west of depot D2, at a similar distance from both. Only trucks from these depots are allowed to
load product at plant Alt1. Thus, the distribution capacity is given by 14 trucks, 8 for LIN and 6 for LOX.
Both 28 LIN customers and 22 LOX customers with varying consumption profiles require inventory
replenishment during a time horizon of one week. Figure 12 shows the plant, depot, alternative source,
and customer locations for the entire supply-chain. Overall, it includes 3 plants, 3 depots, 1 alternative
source, and 50 customers. All problem data for Example 2 are provided as Supplementary Information.
Similar to Example 1, for every plant and customer we assume that the inventory levels at the end of the
time horizon must be at least the same than when the time horizon begins. The overall forecasted product
to be replenished is 50,896 Mcf for LIN and 28,059 Mcf for LOX.
32
Figure 11. Production rate limits for the alternative modes and plants of Example 2.
Figure 12. Supply-chain map for Example 2.
33
Route distances are calculated by using the straight line distance for any pair of locations. The route
generation procedure described in Appendix A is used to propose a sufficiently large route set S. The
parameters of the algorithm and size of the set of routes obtained for each product are shown in Table 11.
Overall, 245 customer sets and 505 alternative routes are proposed. For conciseness we do not report
results for other route sets, although it is clear that changing the parameters in Table 11 can impact the
routes available for the model, and thus the quality of the distribution schedule found.
The model statistics and computational results considering a CPU time limit of 1 h. are shown in Table
12. The model features good computational performance taking into account the model size and the
number of possible routes being tested. On one hand, Table 12 shows that the relaxed solution is close to
the best MIP solution, which clearly indicates that the proposed MILP model has a tight relaxation. On
the other hand, due to the model size and complexity, the convergence rate of the bounds is quite slow
and the best possible solution is nearly midway the relaxed and the MIP solution even after 1 h. of CPU
time. However, taking into account the authors’ experience, a solution with a relative gap of ~2.5% is
excellent for the problem being solved. For instance, while limited by the set of routes proposed, the best
solution cannot improve more than $2,800. At the best solution found, plant P1 produces and sources a
total of 26,962 Mcf of liquid product (LIN + LOX), while plants P2 and P3 produce and source 26,664
Mcf and 25,329 Mcf, respectively.
5.2.1 Impact of electricity price variations
An alternative scenario (A) is introduced to further show the impact of using a coordinated model that
simultaneously takes into account production and distribution decisions. In this case, the electricity prices
(for all time periods, t) are increased by 1 cent for plant P2 and decreased by 0.5 cents for plants P1 and P3.
Example 2 is solved again with the computational results also shown in Table 12. The model features the
same size reported previously. The best solution found decreases from $109,841 to $107,756 with the
modified electricity prices. While the difference between both solutions is small, the impact that the
change of electricity cost has on the selection of production and distribution activities throughout the
entire supply-chain is significant. Figure 13 shows how the total cost of each production and distribution
facility changes between both solutions, and Figure 14 presents a comparison of the product being
sourced from each plant. As it can be seen, the production in Plant P2 decreases, while the production in
P1 and P3 increases due to the changes in electricity prices. The distribution costs of the three depots are
similarly changed.
34
Table 11. Route generation parameters and statistics for Example 2.
LIN LOX Parameters cmax 3 3 dmax 500 500 smax 60 45 vmin 2 2 vmax 5 5 # of customer sets 140 105 # of routes 286 219
Table 12. Computational results for Example 2.
Example 2
Example 2 (A) change of
electricity prices
Example 2 (B) shut-down of plant P2
starting at time t3
Binary variables 13,832 13,832 13,808 Continuous vars. 21,533 21,533 21,533 Constraints 19,993 19,993 19,993 Relaxed LP sol. 104,070 101,032 115,392 MIP solution 109,841 107,756 123,135 Best possible sol. 107,061 104,451 118,505 Rel. gap 2.5% 3.1% 3.7% CPU time 3,600 s 3,600 s 3,600 s Nodes 135,991 183,854 69,979
Figure 13. Comparison of total cost at each production and distribution site for the best solution of
Example 2 considering alternative electricity prices.
35
Figure 14. Product sourced per plant for each scenario of Example 2.
In order to quantify the impact of the aforementioned change of electricity prices, we take the best
solution obtained with the original forecast prices and calculate the production cost of each plant P1 to P3,
but using the modified electricity prices instead. Based on the total product sourced from each plant and
because usppmi is constant, it is possible to derive from the objective function (30) that the additional cost
for plant P2 is $5,332.8, while the cost reduction for plants P1 and P3 is $2,696.2 and $2,532.9,
respectively. Using these results to obtain the production cost of each plant (see Figure 13), both the total
production cost and the simultaneous production and distribution costs are 2% higher than the solution
with the modified prices. Conversely, the same can be observed with the production cost of P1 to P3 of the
best solution for the modified electricity price scenario by using the original forecasted prices instead.
5.2.2 Production capacity disruptions
A second scenario (B) is also considered, this time assuming a shut-down is required for maintenance at
plant P2. The maintenance starts at time t3 (start of second day) and lasts until the end of the week. To
model the shut-down, the RHS of Eqn (1) is set to zero for plant P2 at all time periods t3-t14. The same
route set is used and the computational results are also shown in Table 12. The best solution features a
total cost of $123,135, which is 12% higher than the best solution of the original example. The total
volume sourced from each plant is also included in Figure 14. This scenario requires product to be
purchased from alternative source Alt1 in order to ensure that customer demands are satisfied. However,
as shown in Figure 14, almost all deliveries come from plants P1 and P3. By considering a plant shut-
down, Example 2 (B) illustrates a possible situation in industrial gases supply-chains in which the
proposed computational tool can help to optimally re-organize production and distribution decisions.
36
5.3 Application of the proposed method to industrial size test cases
The proposed simultaneous production and distribution model has been applied to several real test
cases involving the current supply-chain of Air Liquide, a multinational industrial gases company with
operations in 70 countries. The examples include 4 to 15 plants, hundreds of customers, and more than
1000 alternative proposed routes. Because of the problem size, in some examples additional methods such
as clustering and assumptions such as planned deliveries were incorporated in the model to reduce the
complexity of the routing alternatives (see Appendix B). The mathematical model has been applied to
several industrial-size test cases, including both historical and future scenarios. When dealing with
historical test cases, some model variables were fixed based on historical data (plant withdrawals, for
example). Both historical and a fully-coordinated mathematical models were solved and the results
compared. Potential savings around 9% of the total historical cost were identified due to better
production-distribution coordination.
In order to illustrate the complexity of the test cases considered, for medium size examples similar
to Example 2, the model features good computational performances by finding solutions with a relative
gap of ~2% in one hour of CPU time. When large examples are considered (100+ customers) the
computational performance decreases, although the model is still able to find good quality solutions in
reasonable CPU times. As an example, Table 13 includes the problem size, model size, and computational
statistics of a large example related to a segregated market region. The example features a planned
delivery forecast scenario (i.e. customer inventory constraints are not included) with 168 customers and
282 planned deliveries. After applying the route generation procedure, with a total of 1235 alternative
routes the model requires 11,053 binary variables (routes are not available for every time t). Although the
model size is large, a realistic and good quality solution with a 3.6% gap was obtained after 5 h. of CPU
time. The best solution obtained is composed of 48.5% production cost and 51.5% distribution cost. Out
of the 235 trips needed for product distribution, 171 trips (~73%) feature a truck filling ratio (truck load /
truck capacity) higher than 95%. In addition, only four trailers visit an alternative source to purchase
additional product, which amounts to approximately 1% of the total cost.
To improve the model accuracy at the distribution side, traveling distances were obtained using
geographic information system (GIS) software. An efficient implementation of the route generation
method described in Appendix A allowed exploring several thousands of potential candidate routes,
depending on the selected parameters. The algorithm enumerates possible routes by traversing a search
tree where each node represents a route with a given customer set. New nodes are created by adding an
extra customer to each parent node. Time windows, filling ratios and traveling distances are considered
when appropriate to select or reject possible routes. While there is always a correlation between the
37
number of routes proposed and the difficulty to converge to an optimal solution, testing alternative sets of
routes clearly demonstrates the relevance of an appropriate route selection to decrease the distribution
cost.
Table 13. Statistics for an industrial size production-distribution coordination test case featuring planned
deliveries.
Problem size Time periods 14 Plants 4 Products 2 Prod. modes 1 or 2 Alt. sources 4 Depots 4 Trucks 32 Customers 168 Planned deliveries 282 Customer sets 440 Routes 1235 Model size Binary vars. 11,053
Cont. vars 22,086 Constraints 16,243
CPU performance
CPU time 5 h Rel. gap 3.6% Nodes 144,178
6. Conclusions and Future Work
This paper has presented an MILP formulation for the simultaneous coordination of production and
distribution decisions on industrial gases supply-chains. On the production side, the model accounts for
multiple plants running various production modes while producing one or more products. Because air
separation is an energy intensive process, the main component of the production cost is the cost of
electricity, and thus the operation of each plant follows electricity market conditions. On the distribution
side, a combined vehicle routing and inventory management problem, known as an inventory routing
problem (IRP), is considered. The vendor is responsible for inventory replenishments so that customers
do not run out of product. Since the entire supply-chain is included, the IRP considered here includes
multiple products, and multiple sources for each product. A forecast of customer consumption is given to
solve the problem. Trucks departing from depots (located or not at plants) are used to deliver product
from a given source to one or more customers. A route is given by the specification of a depot, a plant,
38
and a customer set to which the product is delivered. Because hundreds of customers are considered, the
number of possible routes grows exponentially. To handle this complexity, the model selects the routes to
be used from a set of proposed routes. Alternative routes for this set are chosen by a pre-processing route
generation algorithm, which inspects thousands of feasible routes taking into account alternative
parameters and a sorting criterion. Overall, the fully-coordinated model includes production decisions at
multiple plants, and distribution decisions at multiple depots.
To asses the impact of a better coordination, different levels of production-distribution coordination were
proposed. While the fully-coordinated model combines dynamic sourcing (ability to serve the same client
from multiple plants) with simultaneous production and distribution, alternatives taken into account
include: (a) either one or multiple plants per customer (fixed sourcing vs. dynamic sourcing), and (b)
either a sequential (production before distribution) or a simultaneous (production and distribution
together) approach. As was shown in Example 1, the capability of the model to perform simultaneous
optimization yields significant cost savings, in both the fixed and dynamic sourcing cases.
The proposed model has been successfully illustrated with a small and a medium size test case, showing
both the capabilities of the model, as well as its computational efficiency that is due to a tight MILP
formulation. The latter allows to readily explore different scenarios such as changes in pricing of
electricity or disruption in the plant operations, as was illustrated in Example 2. Finally, the application to
industrial case studies was discussed in which despite longer computational times, savings of the order of
9% were identified. As for future directions, two areas that deserve attention are the use of decomposition
techniques for reducing the computational times in large industrial problems, and addressing the
uncertainty of model parameters.
Acknowledgments
Financial support from the Center for Advanced Process Decision-making (CAPD) at Carnegie Mellon
University and American Air Liquide is greatly appreciated.
Appendix A - General framework for generating a list of feasible routes
The mathematical model presented in this paper requires a set of alternative routes given as input. This set
represents the possible routes that can be selected by the model to obtain a feasible solution. While it is
possible to enumerate all of them, the number of routes grows exponentially with the number of depots,
plants, and customers. Therefore, it is convenient to reduce the alternatives by filtering out those routes
that are more unlikely to be part of the optimal distribution schedule. By limiting the set of routes
proposed, the size of the model and the computational effort required to find its best solution both
39
decrease. However, this approach can potentially cut off some of the routes needed to obtain an optimal
solution, and thus the optimality of the proposed model is limited by the quality of the set of routes
proposed.
A general framework to generate a set of routes based on the data of depots, plants and customers is
described here. As mentioned in Section 4.2, each route is defined by a tuple (d, p, s), where d is a depot,
p is a plant (source) and s is a set of customers. We assume here that plant, depot, and customer locations
are given and it is possible to calculate the traveling distance between any pair of them. Thus, given a
tuple (d, p, s) it is also possible to calculate the distance disdps for the shortest path to deliver product from
plant p to the customers of s using a truck from depot d.
The procedure ROUTEGENERATION is presented in Table A1. The main parameters of the proposed
method, which can vary for each plant p and product i, are:
• cmax: maximum number of customers visited in a trip,
• dmax: maximum distance for the shortest path of the route,
• smax: maximum number of routes,
• vmin: required number of routes for each customer c and time t∈Tc, and
• vmax: limit on the number of routes for each customer c and time t∈Tc.
The proposed ROUTEGENERATION procedure iterates over all combinations of plants and products adding
the routes obtained to a list of routes R. At each iteration (i.e. for a given plant p and product i), all
possible routes subject to a limit cmax (a given positive integer) on the number of visited customers are
inspected. Customer sets s are generated as combinations (subsets) of n = 1, 2, …, cmax elements taken
from Ci. After inspecting the possible routes, GENERATEFEASIBLEROUTES returns a set with all the tuples
r = (d, p, s) that verify the following conditions: (a) trucks from depot d can source from plant p (p∈Pdi),
(b) all customers of s can receive product from plant p (p∈Ps, where Ps ≠ ∅ as required in Section 4.2.7),
(c) the customer set s verifies ≤s cmax, and (d) the TSP distance of route r does not exceed the limit
(disdps ≤ dmax). Additional conditions can be imposed so that the number of feasible routes does not
become too large. Once all feasible routes are obtained, the resulting set FR is sorted based on a criterion
selected beforehand. To implement the SORT procedure, both the route distance and the logistics ratio (i.e.
cost per volume sourced) were the alternatives evaluated to quantify the convenience of selecting a given
route. The logistics ratio, generating the most economically convenient routes, was used in the test cases.
It is calculated using the maximum volume that can be delivered to the customer set s in a given time t.
Given the sorted list of routes FR, two selection stages are executed to choose the routes required by the
model. SELECTMIN ensures the selection of at least vmin routes for each customer c and time period t∈Tc
when a delivery can be made to this customer. Only if the set of feasible routes FR does not include
40
enough alternatives, vmin different routes are not found. SELECTMAX completes the selection of routes
seeking at least vmax routes for each customer c and time period t∈Tc. However, it finishes earlier
whenever the number of selected routes reaches the maximum quantity smax. The procedure TESTROUTE
is as an auxiliary procedure used for both selection methods. The list of routes R is returned by the
algorithm, from which customer sets Sdi are derived.
Table A1: Route Generation procedure.
procedure ROUTEGENERATION input: {integers} cmax, dmax, smax, vmin, vmax output: {route-list} R begin R ← [] {empty list} for each p∈P, i∈Ip FR ← GENERATEFEASIBLEROUTES(p, i, cmax, dmax) SORT(FR) SELECTMIN(FR, R, vmin) SELECTMAX(FR, R, smax, vmax) end for return R end
procedure SELECTMIN input: {route-list} FR, R, {integer} vmin output: {route-list} FR, R begin i ← 1 while i ≤ size(FR) r ← FR(i) if TESTROUTE(r, vmin) then Select route r Add r to list R end if i ← i + 1 end while end
procedure SELECTMAX input: {route-list} FR, R, {integer} smax, vmax output: {route-list} FR, R begin i ← 1 while (i ≤ size(FR)) and (# of selected routes in FR < smax) r ← FR(i) if TESTROUTE(r, vmax) then Select route r Add r to list R end if i ← i + 1 end while end
41
procedure TESTROUTE input: {route} r, {integer} limit output: {boolean: whether to select a route or not} begin if r is not selected and ∃ customer c and time t such that: (c can receive a delivery at time t using route r and # of selected routes for c at time t < limit) then return true else return false end
Appendix B - Large scale problems
In this section two alternatives to reduce the complexity of the distribution side problem by incorporating
additional assumptions are described: clustering and planned deliveries. These methods can be used,
either solely or combined, to facilitate the solution of industrial size problems otherwise limited by the
computational effort needed to solve a large MILP model.
Clustering methods may be used to reduce the model size when a given problem instance includes
hundreds of customers, which leads to a large increase of the number of alternative routes. Let q∈Q be a
group or cluster of customers and Cq the subset of customers belonging to cluster q. We assume that the
set Q is obtained a priori through the application of some clustering algorithm (Jain et al., 1999) and that
every customer belongs to a unique cluster (at least for each time t.) The location of a cluster q is
calculated as a weighted average of the locations of the customers belonging to q. In turn, the weight cκ
given to a customer c∈Cq is based on an estimation of the minimum number of deliveries necessary to
replenish the consumption of c over the entire time horizon. Eqn (B-1) defines the location ( qx ) of each
cluster and the minimum number of deliveries ( cκ ), where min,avgcQ is the average redline, avg
cR is the
average consumption for customer c∈Ci, and }{maxmax truckkKki UU
i∈= is the capacity of the largest
truck available to deliver product i.
IiCcURQQ
RQq
xx i
iavgc
avgcc
Tt ctc
Cc c
Cc cc
q
q
q ∈∈∀
+−=∈∀
⋅= ∑∑∑
∈
∈
∈ ,},min{
; maxmin,maxκκ
κ (B-1)
Routes distances from a given depot d and plant p are obtained using the location of the cluster given by
Eqn (B-1). An internal distance can be added to each route visiting cluster q, to account for the distance
42
traveled between customers inside the cluster. Two types of routes are considered to deal with clusters on
the distribution side: (i) intra-cluster routes, that only deliver product to all or a subset of the customers Cq
and (ii) inter-cluster routes, where two or more clusters are visited on a given round-trip delivery (i.e. a
truck visits one or more customers of cluster q1 and then one or more customers of cluster q2, etc.)
To handle these alternatives, we extend the routing scheme presented by adding simple conditions when
customer sets are defined. Alternative (i) means that there is only one customer set s for each cluster q.
Moreover, there is a one-to-one correspondence between customer sets s and clusters q. Since it is less
straightforward, alternative (ii) is discussed in more detail. In this case, for every cluster q and customer
set s, either sCq ⊂ or ∅=∩ sCq . In other words, customer sets are defined based on cluster data, so
that each s includes all customers c∈Cq or none of them (an alternative point of view is that each set s
now includes clusters instead of customers).
Given an appropriate definition of the sets s, the variable dsqt is introduced representing the volume
delivered to some or all the customers of q through s. Thus, Eqns (16) and (17) are replaced by the
following constraints:
TtIiSsde isCQqsqt
SsDd Kkkst
qidi di
∈∈∈∀= ∑∑ ∑⊂∈∈∈ ∈
,,)(:)(:
(B-2)
TtIiQqdD isCSs
sqtCc
tcqiq
∈∈∈∀= ∑∑⊂∈∈
,,)(:
, (B-3)
Notice that Eqns (B-2) and (B-3) must be considered together with customer inventory constraints (23)
and (24). No additional changes are introduced in the model, only the cluster locations given by Eqn (B-
1) are used to calculate route distances. Overall, by aggregating customers into clusters the number of
delivery sets s is significantly reduced, which in turn reduces the number of binary variables ykst. The
tradeoff between the accuracy of route distance calculations and the CPU time required to solve the
problem must be evaluated to select between using a detailed customer-based routing approach or an
approximate cluster-based method.
The second alternative to handle large test cases is a reduction of scope of the distribution side problem
by assuming that the amount of product to be delivered to each customer throughout the time horizon is
given. In this case, customer inventory constraints are not needed, and the problem data only specifies the
forecast of planned deliveries instead of the customer consumption profiles. Thus, the distribution side
full inventory routing problem reduces to a smaller vehicle routing problem with time windows (VRP-
43
TW). The complexity of the problem decreases, mainly because the number of routes available at any
given time is restricted by the possible deliveries (open time windows) at that time.
We assume that, for each delivery of product i to customer c, the volume to be delivered and the specific
time window during which the delivery takes place are given. The parameter delivttcU21,, introduced in Section
4.4 is used, where the length of the time intervals (t1, t2) is usually one day. Let Tc be the set of time
periods t when a delivery can be made to customer c, then the accumulated volume of product i that must
be delivered to customer c up to time period t is calculated as:
CcU sumtc ∈∀= 00, (B-4)
0'
,',1,, :, ttTtCcUUUtt
delivttc
sumtc
sumtc >∈∈∀+= ∑
≤− (B-5)
In order to guarantee that the right amount of product is delivered to customer c by the end of each time
window (t1, t2), constraint (B-6) is used. This constraint is defined when *cTt∈ , where
{ }0:'| ,',* >∃∧∈= deliv
ttccc UtTttT includes the upper bound limits of all the time windows of customer c,
and it works properly even if two deliveries have overlapping time windows.
*
,
''
', , csum
tc
ttTt
tc TtCcUDc
∈∈∀=∑≤∈
(B-6)
To handle planned deliveries, customer inventory constraints (23) and (24) are replaced by Equations (B-
4) to (B-6). To use these equations it is important to ensure that, for all model constraints, variables ykst,
ekst, dsct, and dsqt are only defined at time periods such that t∈Ts, where cscs TT ∈= .
As a particular case, if every time window is restricted to a unique period (i.e., t1 = t2), then Equation (B-
6) reduces to Equation (B-7). In this case, the amount of product delivered to customer c at time period t
(Dc,t) becomes a problem parameter.
*
,,, , cdeliv
ttctc TtCcUD ∈∈∀= (B-7)
44
Finally, planned deliveries are particularly useful when combined with clustering methods. For instance,
notice that when definition (B-7) is applied the LHS of Eqn (B-3) can be calculated a-priori.
Nomenclature
Subscripts
c customer d depot i product j product grade k truck m production mode p plant q cluster s customer set (subset of customers visited in a given route)
Sets
C customers Ci customers for product i Cq customers belonging to cluster q D depots I products Ip products of plant p Ipm products produced by plant p while running in mode m Jc product grades that can be delivered to customer c Ji product grades of product i K trucks Kdi trucks for product i available at depot d M production modes Mpt production modes available at plant p in time period t P all plants Palt alternative sources Pown plants owned by the company Pdi plants associated to depot d and product i Pdi,s plants from which a truck from depot d can source product i to customer set s Q clusters of customers Qi clusters of customers for product i Si customer sets for product i Sdi customer sets available for product i and depot d Spi alternative customer sets to source product i from plant p T time periods Tc time periods when a delivery to customer c is possible Ts time periods when a delivery to all customers in s is possible
Parameters
,pm iλα coefficient of the production rate of product i for the limiting hyperplane λ
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dpsδ difference between actual distance disdps and minimum distance mindsdis
max,idpδ maximum dpsδ for all possible sets s∈Sdi
t∆ duration of time period t ),( 21 ttµ total product consumed by customer c in the interval [t1, t2]
pη turndown ratio for plant p ,pm λπ upper bound for the hyperplane λ limiting the feasible rates of production mode m, plant p
cκ estimation of the minimum number of deliveries for customer c initpb whether plant p is running (1) or shut down (0) at time t
ck traveling cost per distance unit for truck k purchase
tpiC , cost of product i if purchased at alternative source p in time t disdps shortest traveling distance of route (d, p, s) obtained by application of a TSP method (customers
of set s are visited using the shortest path starting at plant p and finishing at depot d) mindsdis minimum distance required for a truck of depot d to deliver product from any valid source to
customer set s start
tpF , start-up cost of plant p at time t H time horizon
inicL initial inventory of customer c inipiL initial inventory of product i at plant p minctQ redline (safety stock level) for customer c at time t min
,tpiQ safety stock in time period t for product i at plant p maxcQ storage capacity of customer c maxpiQ storage capacity of product i at plant p purchase
tpiQ , maximum volume of product i available at alternative source p in time t Rct product consumption forecast of customer c at time t
sitetpiR , forecast of gaseous customer pipeline demand for product i at plant p in time period t
delivttcU21,, volume of product required by customer c between time t1 and time t2 (planned delivery)
sumtcU , accumulated volume required by customer c at time t
truckkU trailer capacity for vehicle k withdrawal
tpiU , fixed truck withdrawal volume of product i from plant p at time period t upt electricity price forecast of plant p during time period t usppmi unit specific power
maxpmiw maximum production rate of product i at plant p running production mode m minpmiw minimum production rate of product i at plant p running production mode m
Binary Variables
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startptb denotes that plant p starts operation at time period t
Bpmt denotes that plant p operates in mode m during time period t Ykpt denotes that truck k loads product at plant p in time period t ykst denotes that truck k visits the customers in set s during time period t
Continuous Variables
ktβ additional distance traveled by truck k to load product from a given plant at time t Dc,t total volume delivered to customer c in time period t dsct volume delivered to customer c distributed among customers of set s in time t dsqt volume delivered to cluster q distributed among members of set s in time t
sitetpiD , volume of product i to be gasified and sent by pipeline at plant p in time period t
trucktpiD , volume of product i withdrawn for truck delivery from plant p at time period t
DCostt total distribution cost at time t DISkt distance traveled by truck k at time t Ekpt volume of product withdrawn from plant p and loaded into truck k at time t ekst volume of product delivered by truck k to the customers s in time t Lct inventory of customer c at time t Lpit inventory of product i available at plant p at the end of time period t PCostt total production cost at time t PWp,t power consumption of plant p at time period t Wpmi,t production rate of product i at plant p, when p is running in mode m at time t (zero otherwise).
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Supplementary Information – Problem data for Example 2 Table S1. Inventory at each production facility (Mcf).
P1 P2 P3 LIN LOX LIN LOX LIN LOX Initial 10,000 8,000 8,500 6,500 6,500 7,000 Maximum 18,000 12,000 12,000 9,000 14,000 10,000 Redline 5,000 3,500 3,000 4,000 4,000 3,000
Table S2. Production rate limits for each plant and production mode, given as vertices of the product space (Mcf/h).